Algebraic Topology

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  • Compute the homotopy group π3(S2).
  • Which homotopy classes \alpha\colon {\mathbb P}^2\to {\mathbb P}^2 are there which is the identity on \pi_1({\mathbb P}^2), the fundamental group?
  • What can you say about the homotopy type of the "dunce cap"?
  • Tell us about the Van Kampen theorem. (Stallings)
  • Can you use the Van Kampen theorem to compute the fundamental group of the Hawaiian earring? (Kirby)
  • Is the fundamental group of the Hawaiian earring finite? Free? Countable or Uncountable? (Kirby)
  • Why do you have the Fundamental Theorem of Algebra under algebraic topology in your syllabus? (Stallings)
  • How do you know the fundamental group of S1 is \Z? (Stallings)
  • Find all 2-fold coverings of the figure 8.
  • Find an example of:
    1. Hp(X) = Hp(Y) for all p, but X and Y not homeomorphic.
    2. πp(X) = πp(Y) for all p, but H_*(X)\not=H_*(Y).
  • The example to (2) above seems to contradict Whitehead's Theorem. Do you know why it doesn't contradict it?
  • Compute H_*({\mathbb C}{\mathbf P}^n) and H^*({\mathbb C}{\mathbf P}^n).
  • Define Chern classes and compute them for some examples like {\mathbb C}{\mathbf P}^n.
  • Does "Euler Class" classify all disk bundles over S2?
  • Does C1, the first Chern class, classify all complex line bundles over T2?
  • Compute the intersection form from the framed link which represents the 4-manifold.
  • What is \pi_2(S^2\vee S^2)?
  • Give an example of two spaces which are not homotopy equivalent, but have the same homology.
  • What is π2 of S^2\vee S^1?
  • Calculate π1(X) where X is the three manifold obtained from T^2\times I by identifying the opposite faces by the glueing map (1,0)\mapsto (2,1), (0,1)\mapsto (1,1).
  • Show that the free group on two generators contains the free group on n generators with finite index.
  • Show that every subgroup of a free group is free.
  • Given an example of a pair (X,A) such taht \pi_i(X,A)\ne \pi_i(X/A) for some i.
  • Give an example of two spaces with the same cohomology groups but with a different ring structure.
  • Show that a compact surface with sectional curvature positive everywhere is homeomorphic to S2.
  • Calculate the homology with coefficients in \Z of the Lens space L(a,b).
  • Prove that for any orientable compact 3-manifold M with boundary \partial M that half the first rational homology of \partial M is killed by inclusion into M.
  • Show that an element of Hn − 1 for an orientable n-manifold is represented by a smoothly embedded n − 1-manifold.
  • If a simply-connected CW complex Σ satisfies H_2(\Sigma) = \Z\oplus\Z and Hi(Σ) = 0 for all i\ne 2, then show that Σ is homotopy equivalent to S^2\vee S^2.
  • Show that if G is a finitely generated finitely presented group, then G is the fundamental group of some compact 4-manifold.
  • Show that a simply connected differentiable manifold is orientable.
  • Classify S3 bundles over S5.
  • Show that any two embeddings of a connected closed set X in S2 has homeomorphic complements C1, C2.
  • Show that {\mathbb C}{\mathbb P}^2 does not cover any manifold other than itself.
  • Compute the homology of {\mathbb P}^n. (Stallings)
  • Compute the homology of {\mathbb P}^n with \Z/2\Z. (Stallings)
  • Compute the cohomology of {\mathbb P}^n. (Stallings)
  • Use intersection theory to compute the cup structure of the cohomology of {\mathbb P}^n with \Z/2\Z coefficients where n is odd. (Stallings)
  • What are all of the n-fold covers of the genus 2 surface. (Stallings)
  • What is an H-space? What special property does π1 of an H-space have? Prove it. (Casson)
  • Why can't S2 be an H-space? (Stallings)
  • What is the homology of S^2\times S^2? Cohomology? How is the cohomology related to the homology? What is the cup product structure? (Casson)
  • Suppose that X and Y are simply-connected cell spaces which have the same homology groups. Do they necessarily have the same homotopy groups? Are they necessarily homotopy equivalent? (Givental)
  • Why does Hurwitz's theorem fail for non-simply connected spaces? Give an example of a space X where the action of π1(X) on the higher homotopy groups is not trivial. (Weinstein)
  • What is the Thom class? Let E be the universal bundle over BU(n) and consider F = {\mathbb P}(E \oplus {\mathbb C}). What is the Thom class of the normal bundle to the zero section in F? (Givental)

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